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Calculating the residue of a complex function

  1. Apr 12, 2012 #1
    1. The problem statement, all variables and given/known data

    calculate the residue of the pole at z=i of the function

    f(z)=(1+z^2)^-3

    State the order of the pole

    2. Relevant equations

    I know the residue theorem and also the laurent series expansion but I'm having trouble applying these

    3. The attempt at a solution

    I think the pole is order 3 but I'm having trouble either expanding the function in order to find the a(-1) term or implementing the residue theorem and would appreciate some help! Thanks
     
  2. jcsd
  3. Apr 12, 2012 #2

    micromass

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    Do you know that the residue of a function f at the point a can be calculated by

    [tex]\lim_{z\rightarrow a} (z-a)f(z)[/tex]
     
  4. Apr 12, 2012 #3
    I thought it was dxzktg.png ?
     
  5. Apr 12, 2012 #4

    micromass

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    Yes, of course, sorry. My formula is for a simple pole.
    So, can you use your formula to find the residue?
     
  6. Apr 12, 2012 #5
    Yes but I am having trouble actually implementing the formula and wondered if anyone could do an example to help me understand it?
     
  7. Apr 12, 2012 #6

    micromass

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    Well, first you'll need to determine the order of the pole.

    Your guess is that the order of the pole is 3. So you'll need to look at

    [itex]\lim_{z\rightarrow a} (z-a)^3f(z)[/itex].

    This limit should be nonzero for the order of the pole to be 3. If it is zero, then the pole is of a lower order. If the limit doesn't exist, then the pole is of higher order.

    So, can you calculate that limit?

    Hint: factor the denominator.
     
  8. Apr 13, 2012 #7
    Ah I forgot about factorising, that makes it so much simpler! I get 3/8 I think!
     
  9. Apr 13, 2012 #8
    VERY close. You probably got something like 12*(2i)^(-5), right? Just compute this again. The 3/8 is right, but you are missing two factors.
     
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